Muga G. Time in Quantum Mechanics - Vol. 2
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3 Post-Pauli’s Theorem 41
q
n
p
m
→ T
−m,n
=
1
2
n
n
j=0
n
j
q
j
p
−m
q
n−j
(3.28)
in T
(q, p). In this form, the canonical commutation relation formally reads
[H, T] = iI; this can be shown by using the known algebra of the formal operators
T
−m,n
due to Bender and Dunne [28, 5, 6].
For linear systems, it is clear that the formal time of arrival operator T is obtained
by Weyl quantizing the classical local time of arrival since T
(q, p) = t
0
(q, p).
On the other hand, for non-linear systems, T is the quantization of t
0
(q, p)plus
quantum corrections required to satisfy the time–energy commutation relation; due
to the existence of obstruction to quantization in Euclidean space, T (for non-linear
systems) cannot be constructed, except for linear systems, via direct quantization of
the classical time of arrival.
3.3.7 Example: The Harmonic Oscillator
Let us consider the harmonic oscillator whose interaction potential is given by
V (q) = μω
2
q
2
/2. Substituting the potential back into the general expression for
the classical time of arrival (3.7) yields the time of arrival at the origin T
0
(q, p) =
−tan
−1
(
μωq/ p
)
/ω, for position q and momentum p at t = 0. On the other hand,
the local time of arrival at the origin is given by
t
0
(q, p) =−
∞
k=0
(−1)
k
2k + 1
μ
2k+1
ω
2k
q
2k+1
p
2k+1
, (3.29)
obtained by either expanding T
0
(q, p) or using Eq. (3.27). We demonstrate how
Eq. (3.29) can be obtained using supraquantization.
Substituting the harmonic oscillator potential back into Eq. (3.23), the kernel
factor can be solved to yield [21]
T (q, q
) =
2μω
∞
k=0
1
(2k + 1)!
μω
2
!
2k+1
(q +q
)
2k+1
(q −q
)
2k
=
2μω
sinh
μω
2
(q
2
−q
2
)
(q −q
)
. (3.30)
This solution is unique and analytic in the entire qq
-plane. Using the identity
∞
−∞
y
m−1
sgn(y)e
−ixy
dy = 2(m − 1)!/(i
m
x
m
) , (3.31)
the Wigner transform of the kernel can be calculated to yield
42 E.A. Galapon
T
(q, p) =
1
2i ω
∞
k=0
1
(2k + 1)!
μω
2
!
2k+1
(2q)
2k+1
∞
−∞
v
2k
sgn(v)exp
−i
v p
!
dv
=−
∞
k=0
(−1)
k
2k +1
μ
2k+1
ω
2k
q
2k+1
p
2k+1
. (3.32)
We find that T
(q, p) coincides exactly with the local time of arrival in the neigh-
borhood of the origin.
Now quantizing the local time of arrival yields the harmonic oscillator time of
arrival operator in formal representation-free form
T =−
1
ω
∞
k=0
(−1)
k
2k + 1
μ
2k+1
ω
2k+1
T
−2k−1,2k+1
. (3.33)
This can shown to be formally conjugate with the formal harmonic oscillator Hamil-
tonian
H =
1
2μ
p
2
+
μω
2
2
q
2
. (3.34)
That is, [H, T] = iI on using the formal canonical commutation relation [q, p] =
iI [28]. Here we see that the result of supraquantization coincides with the Weyl
quantization of the local time of arrival.
3.3.8 Example: The Quartic Oscillator
Let us consider the quartic oscillator whose interaction potential is given by V =
λq
4
/4. The local time of arrival can be calculated from Eq. (3.27) to give
t
0
(q, p) =−
∞
j=0
(−1)
j
Γ (5/4)Γ ( j + 1/2)
√
π2
j
Γ ( j +5/4)
μ
j+1
λ
j
q
4 j+1
p
2 j+1
. (3.35)
In the following, we demonstrate that T
(q, p) = t
0
(q, p) +O(
2
). And this is just
a special case of our general result on the equality of T
(q, p) and t
0
(q, p) only in
the limit of vanishing or infinitesimal for non-linear systems.
Substituting the quartic oscillator back into Eq. (3.23), the kernel factor can be
solved to yield [21]
T (q, q
) =
1
4
∞
k=0
∞
j=2k
σ
k, j
μλ
16
2
j−k
(q +q
)
4 j+1−6k
(q −q
)
2 j
, (3.36)
where the σ
k, j
’s are constants given by
3 Post-Pauli’s Theorem 43
σ
0, j
=
Γ (5/4)
8
j
j! Γ ( j +5/4)
, (3.37)
σ
k, j
=
j−2k
m=0
σ
k−1,2(k−1)+m
8
j−2k+1−m
(
−j
)
j−2k+1−m
−
1
4
− j +
3k
2
j−2k+1−m
, (3.38)
for k = 1, 2,... and j ≥ 2k, in which (a)
n
is the Pochammer symbol. This solution
is likewise unique.
Using the same identity, we can similarly calculate the Wigner transform of the
kernel to yield
T
(q, p) =−
∞
j=0
(−1)
j
Γ (5/4)Γ ( j + 1/2)
√
π2
j
Γ ( j +5/4)
μ
j+1
λ
j
q
4 j+1
p
2 j+1
−
2
·
1
4
∞
j=2
(−1)
j
σ
1, j
(2 j)!μ
j
λ
j−1
q
4 j−5
p
2k+1
−···
= τ
0
(q, p) +
2
τ
1
(q, p) +
4
τ
2
(q, p) +··· , (3.39)
where the τ
j
(q, p)’s are independent of . Comparing the leading term, which is
independent of , we find that it is just the local time of arrival at the origin. Thus
the classical time of arrival is recovered from the time of arrival operator.
Quantizing the local time of arrival yields the quartic oscillator time of arrival
operator in formal representation-free form
T =−
∞
j=0
(−1)
j
Γ (5/4)Γ ( j + 1/2)
√
π2
j
Γ ( j +5/4)
μ
j+1
λ
j
T
−4 j−1,2k+1
−
2
·
1
4
∞
j=2
(−1)
j
σ
1, j
(2 j)!μ
j
λ
j−1
T
−4 j+5,2k+1
−···. (3.40)
This can be shown to be formally conjugate with the formal quartic oscillator Hamil-
tonian
H =
1
2μ
p
2
+λq
4
. (3.41)
That is, [H, T] = iI on using the formal canonical commutation relation [q, p] =
iI [28]. We find that the leading term is just the Weyl quantization of the local time
of arrival. However, the appearance of terms in powers of shows that quantization
fails to yield a formal time of arrival operator that is conjugate with the Hamiltonian.
This is a manifestation of the obstruction to quantization in Euclidean space.
44 E.A. Galapon
3.4 Confined Time of Arrival Operators
Technically the confined time of arrival (CTOA) operators are the projections of the
formal operator T in the Hilbert space H
l
= L
2
[−l, l] [24, 25, 22, 28]. Physically
they are the time of arrival operators for a confined particle in the interval [−l, l]
under certain boundary conditions imposed on the momentum operator of the con-
fined particle. Since the time of arrival operator T is explicitly in terms of the posi-
tion and momentum operators, a CTOA operator is the operator T written in terms
of the position and momentum operator of the confined particle. The CTOA operator
is then obtained by specifying the position and momentum operator. The position
operator is unique and is given by the bounded operator q,
(
qϕ
)
(q) = qϕ(q)for
all ϕ(q)inH
l
. On the other hand, the momentum operator is not unique and has to
be considered carefully. We assume the system to be conservative and we require
that the evolution of the system be generated by a purely kinetic Hamiltonian in the
absence of interaction potential. The former requires a self-adjoint Hamiltonian to
ensure that time evolution is unitary. The latter requires a self-adjoint momentum
operator commuting with the kinetic energy operator.
These two requirements are only satisfied by the following choice of the momen-
tum operator. For every
|
γ
|
<π/2, define the self-adjoint momentum opera-
tor (p
γ
φ)(q) =−iφ
(q), with domain D
p
γ
consisting of those vectors φ(q)in
H
l
with square integrable first derivatives and that satisfy the boundary condition
φ(−l) = e
−2iγ
φ(l). With p
γ
self-adjoint, the Hamiltonian is purely kinetic in the
non-interacting case, i.e., H
γ
= (2μ)
−1
p
2
γ
. The momentum and the Hamiltonian
then commute and have the common set of eigenvectors
φ
(γ )
k
(q) =
1
2l
exp
#
i(γ +kπ)
q
l
$
, (3.42)
with respective eigenvalues p
k,γ
= (γ + kπ)l
−1
, E
k
= p
2
k,γ
(2μ)
−1
, for all k = 0,
±1, ±2,.... Since T depends on the momentum operator, the projection of T in H
l
is a family of operators
T
γ
, with each T
γ
corresponding to the momentum p
γ
.
The operators T
γ
are the confined time of arrival operators.
To find the T
γ
’s explicitly we need to have the explicit forms of the operators
T
−m,n
in H
l
for every γ and positive integer m and n, in particular their kernels in
coordinate representation. There are two cases to consider: for non-periodic bound-
ary conditions, γ = 0; and for periodic boundary condition, γ = 0. Both give the
compact and self-adjoint operators
T
γ =0
−m,n
=
1
2
n
j=0
n
j
q
j
p
−m
γ
q
n−j
, T
γ =0
−m,n
=
1
2
n
j=0
n
j
q
j
P
−m
0
q
n−j
, (3.43)
where P
−1
0
= Ep
−1
0
E in which E is the projector onto the subspace orthogonal to
the null space of p
0
. The physical motivation and mathematical justification of using
P
−1
0
is elaborated in [19].
3 Post-Pauli’s Theorem 45
For continuous potentials, the index m is of the form m = 2s + 1forsome
positive integer s. Now in coordinate representation, the operators T
γ
−2s−1,n
in the
Hilbert space H
l
assume the integral form
%
l
−l
q
|
T
γ
−2s−1,n
q
ϕ(q
)dq
, where the
kernels are given by [22]
q
|
T
−2s−1,n
q
=
q + q
2
n
q
|
p
−2s−1
q
, (3.44)
where
q
|
p
−2s−1
γ =0
q
=
1
2sinγ
(−1)
s
(q −q
)
2s
2s+1
(2s)!
e
iγ
H(q −q
) + e
−iγ
H(q
−q)
,(3.45)
q
|
P
−2s−1
0
q
=
i(−1)
s
2
2s+1
(q −q
)
2s
(2s)!
sgn(q −q
) −
i
l
(−1)
s
2
2s+1
(q −q
)
2s+1
(2s + 1)!
. (3.46)
Notice that the second term in Eq. (3.46) can be obtained by integrating the factor of
sgn(q−q
) in the first term with respect to the variable v = (q−q
). This observation
will be important below.
Because the T
γ
−2s−1,n
’s are Fredholm integral operators in coordinate represen-
tation, the projection of the formal operator T in H
l
= L
2
[−l, l] is the family of
Fredholm integral operators
(T
γ
ϕ)(q) =
l
−l
q
|
T
γ
q
ϕ(q
)dq
,
|
γ
|
≤ π/2
. (3.47)
Using the coordinate representation of the operators T
−2s−1,n
in H
l
for a given γ ,
the kernel of T
γ
can be shown to be given by [22]
q
|
T
γ =0
q
=−μ
T (q, q
)
sin γ
e
iγ
H(q −q
)+e
−iγ
H(q
−q)
, (3.48)
q
|
T
γ =0
q
=
μ
i
T (q, q
)sgn(q−q
)−
μ
il
B(q, q
) , (3.49)
where T (q, q
) is the kernel factor and B(q, q
)isgivenby
B(q, q
) =
(q−q
)
0
T (u,v)dv
u=q+q
, (3.50)
in which T (u,v) = T (q, q
) with u = (q + q
) and v = (q − q
). Equation (3.50)
follows from the observation made above concerning Eq. (3.46).
46 E.A. Galapon
For continuous potentials the kernel
q
|
T
γ
q
is square integrable; and since it
is symmetric, the confined time of arrival operator T
γ
is compact and self-adjoint.
4
The compactness of T
γ
implies that T
γ
has a complete set of square integrable
eigenfunctions with a pure discrete spectrum. This further implies that the confined
time of arrival operators cannot be covariant as their eigenvalues do not constitute
the real line. We will see later on that even though the T
γ
’s are non-covariant they
are physically meaningful. Moreover, we will provide for a numerical evidence that
covariance may be recovered in the limit as the confining length l increases indefi-
nitely.
With the kernel factor T (q, q
) = (q + q
)/4 for the free particle, the origin
of the operator −T
2
in Sect. 3.2.4 is now clear: It is just the confined time of
arrival operator for the given system for the given boundary conditions on the
Hamiltonian.
3.5 Conjugacy of the Confined Time of Arrival Operators
We now address the conjugacy of the CTOA operators T
γ
with their respective
Hamiltonians H
γ
. We will show that the commutator domain and the canonical
domain do not coincide and that the confined time of arrival and Hamiltonian pair is
a canonical pair of closed category. The proof of the non-triviality of the canonical
domain is given elsewhere [27].
3.5.1 Non-periodic Boundary Conditions
We first determine the commutator domain D
com
of T
γ
and the Hamiltonian H
γ
for
γ = 0; for continuous potentials, the potential energy operator is defined every-
where in H
l
so that the domain of H
γ
equals the domain of the kinetic energy
operator. Now D
com
consists of all vectors common to the domains of the operators
T
γ
H
γ
and H
γ
T
γ
. Since T
γ
is defined everywhere in the Hilbert space, the domain
of T
γ
H
γ
is just the domain of H
γ
, D
T
γ
H
γ
= D
H
γ
. Since D
H
γ
is dense, D
T
γ
H
γ
is
dense. On the other hand, the domain of H
γ
T
γ
consists of those vectors ϕ in H
such that T
γ
ϕ falls into the domain D
H
γ
of H
γ
. In particular, it consists of those
ϕ(q) such that (T
γ
ϕ)(−l) = e
−2γ
(T
γ
ϕ)(l) and (T
γ
ϕ)
(−l) = e
−2γ
(T
γ
ϕ)
(l).
One can explicitly work out the implementation of these boundary conditions to
yield
4
In [22] two confined time of arrival operators are introduced: The algebra preserving CTOA
operators, which are just the operators we refer to here as the confined time of arrival operators;
and the quantized CTOA operators, which are the projections of the Weyl-quantized local time
of arrival in the Hilbert space H
l
. The two are the same for linear systems, while the later is the
leading term of the former for non-linear systems.
3 Post-Pauli’s Theorem 47
D
H
γ
T
γ
=
ϕ(q) ∈ L
2
[−l, l]: ϕ
(q) ∈ L
2
[−l, l],
l
−l
φ
T
(q
)ϕ(q
)dq
= 0,
l
−l
ϕ
T
(q
)ϕ(q
)dq
+il sin γ
e
−iγ
ϕ(l) +e
iγ
φ(−l)
= 0
, (3.51)
where
φ
T
(q
) =
l
−l
∂T (q, q
)
∂q
dq,ϕ
T
(q
) =
l
−l
∂
2
T (q, q
)
∂q
2
dq . (3.52)
Since D
T
γ
H
γ
= D
H
γ
the domain of the commutator is found by restricting the
domain of H
γ
T
γ
in the domain of H
γ
. Explicitly the commutator domain is
given by
D
[H
γ
,T
γ
]
=
ϕ(q) ∈ D
H
γ
: ϕ
(q) ∈ L
2
[−l, l],
l
−l
φ
T
(q
)ϕ(q
)dq
= 0,
l
−l
ϕ
T
(q
)ϕ(q
)dq
+2il sin γ e
−iγ
ϕ(l) = 0
. (3.53)
Since T (q, q
) has continuous first and second partial derivatives with respect to
both variables, the functions φ
T
(q
) and ϕ
T
(q
) are both square integrable in [−l, l]
so that they belong to H
l
. This implies that the condition
%
l
−l
φ
T
(q
)ϕ(q
)dq
= 0
requires that the vector φ
T
in H
l
be orthogonal to the domain D
H
γ
T
γ
. The commu-
tator domain is then not dense.
For all ϕ in the commutator domain, the commutator can be explicitly evaluated
to yield
[H
γ
, T
γ
]ϕ
(q) = i
&
dT(q, q)
dq
+
∂T (q, q
)
∂q
q
=q
+
∂T (q, q
)
∂q
q
=q
'
ϕ(q)
−
μ
sin γ
l
−l
−
2
2μ
∂T (q, q
)
∂q
2
+
2
2μ
∂T (q, q
)
∂q
2
(V (q) − V (q
))T (q, q
)
×
e
iγ
H(q − q
) +e
−iγ
H(q
−q)
ϕ(q
)dq
+
2sinγ
e
−iγ
ϕ
(l)
T (q, q
)
q
=−l
− T (q, q
)
q
=l
!
−
2sinγ
e
−iγ
ϕ(l)
&
∂T (q, q
)
∂q
q
=−l
−
∂T (q, q
)
∂q
q
=l
'
. (3.54)
We arrive at this expression by performing two successive integration by parts in
evaluating the action of the operator T
γ
H
γ
.
The bracketed quantity in the first term equals one, by virtue of the boundary con-
ditions on the kernel factor Eq. (3.24); and the second term vanishes, since T (q, q
)
solves Eq. (3.23). Then the required commutator is given by
48 E.A. Galapon
[H
γ
, T
γ
]ϕ
(q) = iϕ(q) +
2sinγ
e
−iγ
ϕ
(l)
T (q, q
)
q
=−l
− T (q, q
)
q
=l
!
−
2sinγ
e
−iγ
ϕ(l)
&
∂T (q, q
)
∂q
q
=−l
−
∂T (q, q
)
∂q
q
=l
'
.(3.55)
If the second and third terms vanish, then H
γ
and T
γ
are conjugate in their entire
commutator domain. However, they do not generally vanish because the vectors
ϕ(q)inD
[H
γ
,T
γ
]
, together with their first derivatives, do not necessarily vanish at the
boundaries.
But if the commutator is further restricted to those elements of the commutator
domain that satisfies ϕ(∂) = 0
or ϕ(−l) = ϕ(l) = 0
and ϕ
(∂) = 0
or ϕ
(−l) =
ϕ
(l) = 0
, the pair forms a canonical pair,
[H
γ
, T
γ
]ϕ
(q) = iϕ(q) for all ϕ(q) ∈ D
can
(H
γ
, T
γ
) , (3.56)
D
can
(H
γ
, T
γ
) =
ϕ(q) ∈ D
H
γ
: ϕ(∂) = 0,ϕ
(∂) = 0,
l
−l
φ
T
(q
)ϕ(q
)dq
= 0,
l
−l
ϕ
T
(q
)ϕ(q
)dq
= 0
. (3.57)
The canonical domain is orthogonal to the vectors φ
T
and ϕ
T
, hence not dense. The
pair H
γ
and T
γ
then forms a canonical pair of closed category.
3.5.2 Periodic Boundary Condition
The commutator domain for H
0
and T
0
can be similarly derived. It is given by
D
[H
0
,T
0
]
=
ϕ(q) ∈ D
H
0
:
l
−l
φ
T,B
(q
)ϕ(q
)dq
= 0 ,
l
−l
ϕ
T,B
(q
)ϕ(q
)dq
+4lϕ(l) = 0
, (3.58)
where
φ
T,B
(q
) =
T (q, q
)
q=l
+ T (q, q
)
q=−l
!
−
1
l
B(q, q
)
q=l
− B(q, q
)
q=−l
!
,
(3.59)
3 Post-Pauli’s Theorem 49
ϕ
T,B
(q
) =
&
∂T (q, q
)
∂q
q=l
+
∂T (q, q
)
∂q
q=−l
'
−
1
l
&
∂ B(q, q
)
∂q
q=l
−
∂ B(q, q
)
∂q
q=−l
'
. (3.60)
Since T (q, q
) and B(q, q
) are both continuous, together with their first derivatives,
φ
TB
(q
) and ϕ
TB
(q
) both belong to the Hilbert space. Then because of the condition
%
l
−l
φ
TB
(q
)ϕ(q
)dq
= 0, the commutator domain is not dense.
For all ϕ in the commutator domain, the commutator can be explicitly evaluated
to yield
(
[H
0
, T
0
]ϕ
)
(q) = iϕ(q)
−
μ
il
l
−l
−
2
2μ
∂
2
B(q, q
)
∂q
2
+
2
2μ
∂
2
B(q, q
)
∂q
2
+(V (q) −V (q
))B(q, q
)
ϕ(q
)dq
−
i
2
ϕ
(l)
T (q, q
)
q
=l
+ T (q, q
)
q
=−l
!
+
1
l
B(q, q
)
q
=l
− B(q, q
)
q
=−l
!
+
i
2
ϕ(l)
&
∂T (q, q
)
∂q
q
=l
+
∂T (q, q
)
∂q
q
=−l
'
+
1
l
&
∂ B(q, q
)
∂q
q
=l
−
∂ B(q, q
)
∂q
q
=−l
'
, (3.61)
where we have performed the same simplifications done above to arrive at the first
term. Again the pair is not conjugate in the entire commutator domain. However, we
see that further restriction of the commutator domain yields a canonical domain. In
contrast to the non-periodic case, vanishing of the ϕ(q)’s and their derivatives at the
boundaries is not sufficient to enforce conjugacy of the pair.
Let us define the integral operator (ϕB)(q) =
%
l
−l
q
|
B
q
ϕ(q
)dq
with kernel
given by
q
|
B
q
=−
2
2μ
∂
2
B(q, q
)
∂q
2
+
2
2μ
∂
2
B(q, q
)
∂q
2
+(V (q) − V (q
))B(q, q
) . (3.62)
This does not vanish because B(q, q
) is not a solution to Eq. (3.23). Thus restricting
the commutator domain to those vectors that belong to the null space of B and to
those vectors, together with their first derivative, that vanish at the boundaries yields
a canonical domain. Then we have the canonical commutation relation
(
[H
0
, T
0
]ϕ
)
(q) = iϕ(q) for all ϕ(q) ∈ D
can
(H
0
, T
0
) , (3.63)
50 E.A. Galapon
D
can
(H
0
, T
0
) =
ϕ(q) ∈ D
H
0
: ϕ(∂) = 0,ϕ
(∂) = 0,
l
−l
φ
TB
(q
)ϕ(q
)dq
= 0,
l
−l
ϕ
TB
(q
)ϕ(q
)dq
= 0, (Bϕ)(q) = 0
. (3.64)
The canonical domain is again not dense because there are non-zero vectors orthog-
onal to the canonical domain; moreover, it is contained in the null space of B, which
is not dense. The pair H
0
and T
0
then forms a canonical pair of closed category.
The above results are meaningful only if the canonical domain is not trivial. In
the following, we will consider the non-interacting case to demonstrate the non-
triviality of the canonical domain of the confined time of arrival operators. In the
process, we will be introducing a method of constructing explicitly the vectors of
the canonical domain.
3.5.3 Example: Non-interacting Case
3.5.3.1 Non-periodic Boundary Conditions
For the non-interacting case, the kernel factor is given by T (q, q
) = (q + q
)/4.
From this, the vectors φ
T
and ϕ
T
orthogonal to the canonical domain can be deter-
mined to yield φ
T
(q
) = l/2 and ϕ
T
(q
) = 0. The canonical domain reduces to
D
can
(H
γ
, T
γ
) =
ϕ(q) ∈ D
H
0
: ϕ(∂) = 0,ϕ
(∂) = 0,
l
−l
ϕ(q)dq = 0
. (3.65)
Clearly D
can
is not dense because it is orthogonal to the non-zero vector φ(q) =
constant. We now show that this subspace is not trivial and in fact it contains an
infinite number of elements.
We do so by constructing explicit elements of the canonical domain. Every
element of D
H
γ
can be expanded in terms of the complete eigenfunctions of the
momentum operator, ϕ(q) =
k
ϕ
k
φ
(γ )
k
(q), where ϕ
k
=
%
l
−l
φ
(γ )∗
k
(q) ϕ(q)dq.For
ϕ(q) to belong to the domain of the Hamiltonian, it must have a square integrable
second derivative; this imposes the condition
k
k
4
|
ϕ
k
|
2
< ∞ on the ϕ
k
’s. Now
ϕ(q) belongs to the canonical domain if it satisfies all the required conditions, which
translates to conditions on the coefficients ϕ
k
’s. The conditions ϕ(∂) = 0, ϕ
(∂) = 0,
and
%
l
−l
ϕ(q)dq translate respectively into
∞
k=−∞
(−1)
k
ϕ
k
= 0,
∞
k=−∞
(−1)
k
k ϕ
k
= 0,
∞
k=−∞
(−1)
k
kπ + γ
ϕ
k
= 0 . (3.66)
For vectors in the domain of the Hamiltonian, Eqs. (3.66) determine a system of
three equations determining three of the ϕ
k
’s in terms of the rest of the coefficients.
This allows us to construct the basic vectors that belong to the canonical domain,